Auxiliary particle filtering: recent developments Nick Whiteley and Adam M. Johansen Summarized by Eun-Sol Kim

Background -SSMs(State Space Models) x1 f x2 … xn g y1 y2 … yn

Background -Particle Filtering The integrals required for a Bayesian recursive filter cannot solved analytically *Prediction step *Update step So, we represent the posterior probabilities by a set of randomly chosen weighted samples

Background - Sequential Importance Resampling

Weaknesses of SIR If there is an outlier, SIR is not robust. If the observation density is tailed distribution, SIR is not robust From Filtering via Simulation: Auxiliary Particle Filters (1999, M.K.Pitt & N. Shephard)

Main idea of APF Auxiliary variable(particle index): k

Algorithm for APF

Generic approaches choosing predictive likelihood Using the approximations of the transition densities and update densities The multivariate t distribution (centered at the mode) In the multimodal case, a mixture of multivariate t distributions.

Experiment Compare the performance of the PF and APF for an angular time series model Hidden states 𝛼 𝑡 =( 𝑥 𝑡 , 𝑣𝑥 𝑡 , 𝑧 𝑡 , 𝑣𝑧 𝑡 )′

Experimental results (1/2)

Experimental results (2/2)